Galois module structure of Milnor K-theory in characteristic p

Abstract

Let E be a cyclic extension of degree p^n of a field F of characteristic p. Using arithmetic invariants of E/F we determine k_mE, the Milnor K-groups K_mE modulo p, as Fp[Gal(E/F)]-modules for all m in N. In particular, we show that each indecomposable summand of k_mE has Fp-dimension a power of p. That all powers p^i, i=0,1,...,n, occur for suitable examples is shown in a subsequent paper [MSS2], where additionally the main result of this paper becomes an essential induction step in the determination of K_mE/p^sK_mE as (Z/p^sZ)[Gal(E/F)]-modules for all m, s in N.

arXiv commentv5 (10 pages); revised introduction and shortened paper; incorporated and acknowledged suggestions by readers of earlier versions

References in corpus (1)

• Galois module structure of pth-power classes of cyclic extensions of degree p^n

Cited by in corpus (6)

• Galois module structure of square power classes for biquadratic extensions
• Galois module structure of Milnor K-theory mod p^s in characteristic p
• On the indecomposability of a remarkable new family of modules appearing in Galois theory
• Arithmetic properties encoded in the Galois module structure of $K^{\times }/K^{\times p^m}$
• Absolute Galois groups viewed from small quotients and the Bloch-Kato conjecture
• Galois module structure of Galois cohomology and partial Euler-Poincare characteristics